PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
BENJAMIN MATSCHKE, JULIAN PFEIFLE, AND VINCENT PILAUD
Abstract. We introduce PSN polytopes whose k-skeleton is combinatorially equivalent to
that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes.
We construct PSN polytopes by three different methods, the most versatile of which is an
extension of Sanyal & Ziegler’s “projecting deformed products” construction to products of
arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k +r +1.
Using topological obstructions similar to those introduced by Sanyal to bound the number
of vertices of Minkowski sums, we show that this dimension is minimal if we moreover require
the PSN polytope to be obtained as a projection of a polytope combinatorially equivalent to
the product of r simplices, when the sum of their dimensions is at least 2k.
1. Introduction
1.1. Definitions. Let 4n denote the n-dimensional simplex. For any tuple n = (n1 , . . . , nr )
of integers, we denote
P by 4n the product of simplices 4n1 × · · · × 4nr . This is a polytope of dimension
ni , whose non-empty faces are obtained as products of non-empty faces
of the simplices 4n1 , . . . , 4nr . For example, Figure 1 represents the graphs of 4i × 46 ,
for i ∈ {1, 2, 3}.
Figure 1. The graphs of the products 4(i,6) = 4i × 46 , for i ∈ {1, 2, 3}.
We are interested in polytopes with the same “initial” structure as these products.
Definition 1.1. Let k ≥ 0 and n = (n1 , . . . , nr ), with r ≥ 1 and ni ≥ 1 for all i. A polytope
is (k, n)-prodsimplicial-neighborly — or (k, n)-PSN for short — if its k-skeleton is combinatorially equivalent to that of 4n = 4n1 × · · · × 4nr .
Benjamin Matschke was supported by DFG research group Polyhedral Surfaces and by Deutsche Telekom
Stiftung. Julian Pfeifle was supported by grants MTM2006-01267 and MTM2008-03020 from the Spanish
Ministry of Education and Science and 2009SGR1040 from the Generalitat de Catalunya. Vincent Pilaud was
partially supported by grant MTM2008-04699-C03-02 of the Spanish Ministry of Education and Science.
1
2
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
This definition is essentially motivated by two particular classes of PSN polytopes:
(1) neighborly polytopes arise when r = 1;
(2) neighborly cubical polytopes [JS07, SZ09] arise when n = (1, 1, . . . , 1).
Remark 1.2. In the literature, a polytope is k-neighborly if any subset of at most k of its
vertices forms a face. Observe that such a polytope is (k − 1, n)-PSN with our notation.
P
ni . We are
Obviously, the product 4n itself is a (k, n)-PSN polytope of dimension
naturally interested in finding (k, n)-PSN polytopes in smaller dimensions. For example,
the cyclic polytope C2k+2 (n + 1) is a (k, n)-PSN polytope in dimension 2k + 2. We denote
by δ(k, n) the smallest possible dimension that a (k, n)-PSN polytope can have.
PSN polytopes can be obtained by projecting the product 4n , or a combinatorially equivalent polytope, onto a smaller subspace. For example, the cyclic polytope C2k+2 (n + 1) (just
like any polytope with n + 1 vertices) can be seen as a projection of the simplex 4n to R2k+2 .
Definition 1.3. A (k, n)-PSN polytope is (k, n)-projected-prodsimplicial-neighborly — or
(k, n)-PPSN for short — if it is a projection of a polytope combinatorially equivalent to 4n .
We denote by δpr (k, n) the smallest possible dimension of a (k, n)-PPSN polytope.
1.2. Outline and main results. The present paper may be naturally divided into two parts.
In the first part, we present three methods for constructing low-dimensional PPSN polytopes:
(1) Reflections of cyclic polytopes;
(2) Minkowski sums of cyclic polytopes;
(3) Deformed Product constructions in the spirit of Sanyal & Ziegler [Zie04, SZ09].
The second part derives topological obstructions for the existence of such objects, using
techniques developed by Sanyal in [San09] (see also [RS09]) to bound the number of vertices
of Minkowski sums. In view of these obstructions, our constructions in the first part turn out
to be optimal for a wide range of parameters.
Constructions. Our first non-trivial example is a (k, (1, n))-PSN polytope in dimension
2k + 2, obtained by reflecting the cyclic polytope C2k+2 (n + 1) in a well-chosen hyperplane:
Proposition 2.3. For any k ≥ 0, n ≥ 2k + 2 and λ ∈ R sufficiently large, the polytope
2k+1
2k+2 T
T
P := conv (ti , . . . , t2k+2
)
|
i
∈
[n
+
1]
∪
(t
,
.
.
.
,
t
,
λ
−
t
)
|
i
∈
[n
+
1]
i
i
i
i
is a (k, (1, n))-PSN polytope of dimension 2k + 2.
For example, this provides us with a 4-dimensional polytope whose graph is the cartesian
product K2 × Kn , for any n ≥ 3.
Next, forming a well-chosen Minkowski sum of cyclic polytopes yields explicit coordinates
for (k, n)-PPSN polytopes:
Theorem 2.6. Let k ≥ 0 and n = (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 1 for all i. There exist
index sets I1 , . . . , Ir ⊂ R, with |Ii | = ni for all i, such that the polytope
P := conv{wa1 ,...,ar | (a1 , . . . , ar ) ∈ I1 × · · · × Ir } ⊂ R2k+r+1
T
P
P
is (k, n)-PPSN, where wa1 ,...,ar := a1 , . . . , ar , i∈[r] a2i , . . . , i∈[r] a2k+2
. Consequently,
i
δ(k, n) ≤ δpr (k, n) ≤ 2k + r + 1.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
3
For r = 1 we recover neighborly polytopes.
Finally, we extend Sanyal & Ziegler’s technique of “projecting deformed products of polygons” [Zie04, SZ09] to products of arbitrary simple polytopes: we suitably project a suitable
polytope combinatorially equivalent to a given product of simple polytopes in such a way
as to preserve its complete k-skeleton. More concretely, we describe how to use colorings of
the graphs of the polar polytopes of the factors in the product to raise the dimension of the
preserved skeleton. The basic version of this technique yields the following result:
Proposition 3.4. Let P1 , . . . , Pr be simple polytopes. For each polytope Pi , denote by ni
its dimension, by mi its number of facets, and by χi := χ(sk1 Pi4 ) the chromatic number of
the graph of its polar polytope Pi4 . For a fixed integer d ≤ n, let t be maximal such that
Pt
i=1 ni ≤ d. Then there exists a d-dimensional polytope whose k-skeleton is combinatorially
equivalent to that of the product P1 × · · · × Pr as soon as
!%
$
t
r
t
X
X
X
1
d−1+
0 ≤ k ≤
(χi − ni )
.
(ni − mi ) +
(mi − χi ) +
2
i=1
i=1
i=1
A family of polytopes that minimize the last summand are products of even polytopes (all
2-dimensional faces have an even number of vertices). See Example 3.5 for the details, and
the end of Section 3.1 for extensions of this technique.
Specializing the factors to simplices provides another construction of PPSN polytopes.
When some of these simplices are small compared to k, this technique in fact yields our best
examples of PPSN polytopes:
Theorem 3.8. For any k ≥ 0 and n = (n1 , . . . , nr ) with 1 = n1 = · · · = ns < ns+1 ≤ · · · ≤ nr ,


2(k + r) − s − t if 3s ≤ 2k + 2r,
δpr (k, n) ≤
2(k + r − s) + 1 if 3s = 2k + 2r + 1,


2(k + r − s + 1) if 3s ≥ 2k + 2r + 2,
P
where t ∈ {s, . . . , r} is maximal such that 3s + ti=s+1 (ni + 1) ≤ 2k + 2r.
If ni = 1 for all i, we recover the neighborly cubical polytopes of [SZ09].
Obstructions. In order to derive lower bounds on the minimal dimension δpr (k, n) that
a (k, n)-PPSN polytope can have, we apply and extend a method due to Sanyal [San09].
For any projection which preserves the k-skeleton of 4n , we construct via Gale duality a
simplicial complex guaranted to be embeddable in a certain dimension. The argument is then
a topological obstruction based on Sarkaria’s criterion for the embeddability of a simplicial
complex in terms of colorings of Kneser graphs [Mat03]. We obtain the following result:
Corollary 4.13. Let n = (n1 , . . . , nr ) with 1 = n1 = · · · = ns < ns+1 ≤ · · · ≤ nr . Then
(1) If
r
X
ni − 2
s−1
0 ≤ k ≤
+ max 0,
,
2
2
i=s+1
then δpr (k,
Pn) ≥ 2k + r − s + 1. P
(2) If k ≥ 21 i ni then δpr (k, n) ≥ i ni .
4
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
In particular, the upper and lower bounds provided by Theorem 2.6 and Corollary 4.13
match over a wide range of parameters:
Theorem 1.4.
= (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 2 for all i, and for any k such
P Forany n
that 0 ≤ k ≤ i∈[r] ni2−2 , the smallest (k, n)-PPSN polytope has dimension exactly 2k+r+1.
In other words:
δpr (k, n) = 2k + r + 1.
Remark 1.5. During the final stages of completing this paper, we learned that Rörig and
Sanyal [San08, Rör08, RS09] also applied Sanyal’s topological obstruction method to derive
lower bounds on the target dimension of a projection preserving skeleta of different kind of
products (products of polygons, products of simplices, and wedge products of polytopes).
In particular, for a product ∆n × · · · × ∆n of r identical simplices, r ≥ 2, they obtain our
Theorem 4.9 and a result [RS09, Theorem 4.5] that is only slightly weaker than Theorem 4.12
in this setting (compare with Sections 4.5 and 4.6).
2. Constructions from cyclic polytopes
Let t 7→ µd (t) := (t, t2 , . . . , td )T be the moment curve in Rd , t1 < t2 < · · · < tn be n distinct
real numbers and Cd (n) := conv{µd (ti ) | i ∈ [n]} denote the cyclic polytope in its realization
on the moment curve. We refer to [Zie95, Theorem 0.7] and [dLRS, Corollary 6.1.9] for
combinatorial properties of Cd (n), in particular Gale’s Evenness Criterion which characterizes
the index sets of upper and lower facets of Cd (n).
Cyclic polytopes yield our first examples of PSN polytopes:
Example 2.1. For any integers k ≥ 0 and n ≥ 2k + 2, the cyclic polytope C2k+2 (n + 1) is
(k, n)-PPSN.
Example 2.2. For any k ≥ 0 and n = (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 1 for all i, define
I := {i ∈ [r] | ni ≥ 2k + 3}. Then the product
Y
Y
C2k+2 (ni + 1) ×
4ni
i∈I
i∈I
/
is a (k, n)-PPSN polytope of dimension (2k + 2)|I| +
when I is not empty). Consequently,
P
i∈I
/
ni (which is smaller than
δ(k, n) ≤ δpr (k, n) ≤ (2k + 2)|I| +
X
P
ni
ni .
i∈I
/
2.1. Reflections of cyclic polytopes. Our next example deals with the special case of
the product 41 × 4n of a segment by a simplex. Using products of cyclic polytopes as in
Example 2.2, we can realize the k-skeleton of this polytope in dimension 2k + 3. We can lower
this dimension by 1 by reflecting the cyclic polytope C2k+2 (n+1) in a well-chosen hyperplane:
Proposition 2.3. For any k ≥ 0, n ≥ 2k + 2 and λ ∈ R sufficiently large, the polytope
2k+1
2k+2 T
T
P := conv (ti , . . . , t2k+2
)
|
i
∈
[n
+
1]
∪
(t
,
.
.
.
,
t
,
λ
−
t
)
|
i
∈
[n
+
1]
i
i
i
i
is a (k, (1, n))-PSN polytope of dimension 2k + 2.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
5
Proof. The polytope P is obtained as the convex hull of two copies of the cyclic polytope
C2k+2 (n + 1). The first one Q := conv{µ2k+2 (ti ) | i ∈ [n + 1]} lies on the moment curve µd ,
while the second one is obtained as a reflection of Q with respect to a hyperplane that is
orthogonal to the last coordinate vector u2k+2 and sufficiently far away. During this process,
(1) we destroy all the faces of Q only contained in upper facets of Q;
(2) we create prisms over faces of Q that lie in at least one upper and one lower facet
of Q. In other words, we create prisms over the faces of Q strictly preserved under
the orthogonal projection π : R2k+2 → R2k+1 with kernel Ru2k+2 .
The projected polytope π(Q) is nothing but the cyclic polytope C2k+1 (n + 1). Since this
polytope is k-neighborly, any (≤ k − 1)-face of Q is strictly preserved by π, and thus, we take
a prism over all (≤ k − 1)-faces of Q.
Thus, in order to complete the proof that the k-skeleton of P is that of 41 × 4n , it is
enough to show that any k-face of Q remains in P . This is obviously the case if this k-face is
also a k-face of C2k+1 (n + 1), and follows from the next combinatorial lemma otherwise. Lemma 2.4. A k-face of C2k+2 (n + 1) which is not a k-face of C2k+1 (n + 1) is only contained
in lower facets of C2k+2 (n + 1).
Proof. Let F ⊂ [n + 1] be a k-face of C2k+2 (n + 1) that is contained in at least one upper
facet G ⊂ [n + 1] of C2k+2 (n + 1). Then,
(1) if n + 1 ∈ G r F , then G r {n + 1} is a facet of C2k+1 (n + 1) containing F .
(2) otherwise, n + 1 ∈ F , and F 0 := F r {n + 1} has only k elements. Thus, F 0 is a face
of C2k (n), and can be completed to a facet of C2k (n). Adding the index n + 1 back
to this facet, we obtain a facet of C2k+1 (n + 1) containing F .
In both cases, we have shown that F is a k-face of C2k+1 (n + 1).
2.2. Minkowski sums of cyclic polytopes. Our next examples are Minkowski sums of
cyclic polytopes. We first describe an easy construction that avoids all technicalities, but
only yields (k, n)-PPSN polytopes in dimension 2k + 2r. After that, we show how to reduce
the dimension to 2k + r + 1, which according to Corollary 4.13 is best possible for large ni ’s.
Proposition 2.5. Let k ≥ 0 and n = (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 1 for all i. For any
pairwise disjoint index sets I1 , . . . , Ir ⊂ R, with |Ii | = ni for all i, the polytope
P := conv va1 ,...,ar | (a1 , . . . , ar ) ∈ I1 × · · · × Ir ⊂ R2k+2r
is (k, n)-PPSN, where
T

va1 ,...,ar := 
X
i∈[r]
ai ,
X
a2i , . . . ,
i∈[r]
X
 ∈ R2k+2r .
a2k+2r
i
i∈[r]
Proof. The vertex set of 4n is indexed by I1 × · · · × Ir . Let A = A1 × · · · × Ar ⊂ I1 × · · · × Ir
define a k-face of 4n . Consider the polynomial
f (t) :=
Y Y
(t − a)2 =
2k+2r
X
cj tj .
j=0
i∈[r] a∈Ai
P
Since A indexes a k-face of 4n , we know that
|Ai | = k + r, so that Sthe degree of f (t)
is indeed 2k + 2r. Since f (t) ≥ 0, and equality holds if and only if t ∈ i∈[r] Ai , the inner
6
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
product (c1 , . . . , c2k+2r ) · va1 ,...,ar equals
 P

i∈[r] ai
X 2k+2r
X
X


..
(c1 , . . . , c2k+2r ) 
=
cj aji =
f (ai ) − c0 ≥ −rc0 ,

.
P
2k+2r
i∈[r] j=1
i∈[r]
i∈[r] ai
with equality
P if and only if (a1 , . . . , ar ) ∈ A. Thus, A indexes a face of P defined by the linear
inequality i∈[r] ci xi ≥ −rc0 .
To realize the k-skeleton of 4n1 × · · · × 4nr even in dimension 2k + r + 1, we slightly modify
this construction in the following way.
Theorem 2.6. Let k ≥ 0 and n = (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 1 for all i. There exist
index sets I1 , . . . , Ir ⊂ R, with |Ii | = ni for all i, such that the polytope
P := conv{wa1 ,...,ar | (a1 , . . . , ar ) ∈ I1 × · · · × Ir } ⊂ R2k+r+1
is (k, n)-PPSN, where
wa1 ,...,ar :=
a1 , . . . , ar ,
X
a2i , . . . ,
i∈[r]
X
a2k+2
i
T
∈ R2k+r+1 .
i∈[r]
Proof. We will choose the index sets Ii to be sufficiently separated in a sense that will be
made explicit later in the proof. This choice will enable us, for each k-face F of 4n indexed
by A1 × · · · × Ar ⊂ I1 × · · · × Ir , to construct a monic polynomial (i.e., a polynomial with
leading coefficient equal to 1)
2k+2
X
fF (t) :=
cj tj
j=0
that has, for all i ∈ [r], the form
fF (t) = Qi (t)
Y
(t − a)2 + si t + ri
a∈Ai
with polynomials Qi (t) everywhere positive and certain reals ri and si . From the coefficients
of this polynomial, we build the vector
nF := (s1 − c1 , . . . , sr − c1 , −c2 , −c3 , . . . , −c2k+2 ) ∈ R2k+r+1 .
To prove that nF is a face-defining normal vector for F , take an arbitrary (a1 , . . . , ar ) in
I1 × · · · × Ir , and consider the following inequality for the inner product:


2k+2
X
X
si ai −
nF · wa1 ,...,ar =
cj aj 
i
j=1
i∈[r]
=
X
(si ai + c0 − fF (ai ))
i∈[r]

=
X
i∈[r]
c0 − Qi (ai )

Y
a∈Ai
(ai − a)2 − ri  ≤ rc0 −
X
ri .
i∈[r]
Equality holds if and only if (a1 , . . . , ar ) ∈ A1 ×· · ·×Ar . Given the existence of a polynomial fF
with the claimed properties, this proves that A1 × · · · × Ar indexes all wa1 ,...,ar ’s that lie on a
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
7
face F 0 in P , and they of course span F 0 by definition of P . To prove that F 0 is combinatorially
equivalent to F it suffices to show that each wa1 ,...,ar ∈ F 0 is in fact a vertex of P , since P
is a projection of 4n . This can be shown with the normal vector (2a1 , . . . , 2ar , −1, 0, . . . , 0),
using the same calculation as before.
Before showing how to choose the index sets Ii that enable us to construct the polynomials fF in general, we would like to make a brief aside to show the smallest example.
Example 2.7. For k = 1 and r = 2, choose the index sets I1 , I2 ⊂ R arbitrarily, but separated
in the sense that the largest element of I1 be smaller than the smallest element of I2 . For any
1-dimensional face F of P indexed by {a, b} × {c} ⊂ I1 × I2 , consider the polynomial fF of
degree 2k + 2 = 4:
fF (t) := (t − a)2 (t − b)2 = (t2 + αt + β)(t − c)2 + s2 t + r2 ,
where
α := 2(−a − b + c),
β := a2 + b2 + 3c2 + 4ab − 4ac − 4bc,
r2 := a2 b2 − βc2 ,
s2 := −2a2 b − 2ab2 − αc2 + 2βc.
Since the index sets I1 , I2 are separated, the discriminant α2 − 4β = −8(c − a)(c − b) is
negative, which implies that the polynomial Q2 (t) = t2 + αt + β is positive for all values of t.
Proof of Theorem 2.6, continued. We still need to show how to choose the index sets Ii that
enable us to construct the polynomials fF in general. Once we have chosen these index sets,
finding fF is equivalent to the task of finding polynomials Qi (t) such that
(i) Qi (t) is monic of degree 2k + 2 −Q
2|Ai |.
(ii) The r polynomials fi (t) := Qi (t) a∈Ai (t − a)2 are equal up to possibly the coefficients
in front of t0 and t1 .
(iii) Qi (t) > 0 for all t ∈ R.
The first two items form a linear equation system on the coefficients of the Qi (t)’s which
has the same number of equations as variables. We will show that it has a unique solution if
one chooses the right index sets Ii (the third item will be dealt with at the end). To do this,
choose pairwise distinct reals ā1 , . . . , ār ∈ R and look at the similar equation system:
(i) Q̄i (t) are monic polynomials of degree 2k + 2 − 2|Ai |.
(ii) The r polynomials f¯i (t) := Q̄i (t)(t − āi )2|Ai | are equal up to possibly the coefficients in
front of t0 and t1 .
The first equation system moves into the second when we deform the points of the sets Ai
continuously to āi , respectively. If we show that the second equation system has a unique
solution then so has the first equation system as long as we have chosen the sets Ii close
enough to the āi ’s, by continuity of the determinant (note that in the end, we can fulfill all
these closeness conditions required for all k-faces of 4n since there are only finitely many
k-faces).
Note that a polynomial f¯i (t) of degree 2k + 2 has the form
(1)
Q̄i (t)(t − āi )2|Ai | + si t + ri
for a monic polynomial Q̄i and some reals si and ri , if and only if f¯i00 (t) has the form
(2)
Ri (t)(t − āi )2(|Ai |−1)
8
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
for a polynomial Ri (t) with leading coefficient (2k + 2)(2k + 1). The backward direction can
be settled by assuming without loss of generality āi = 0 (otherwise just make a variable shift
(t − āi ) 7→ t) and then integrating (2) twice with integration constants zero to obtain (1).
Therefore the second equation system is equivalent to the following third one:
(i) Ri (t) are polynomials of degree 2k − 2(|Ai | − 1) with leading coefficient (2k + 2)(2k + 1).
(ii) The r polynomials gi (t) := Ri (t) · (t − āi )2(|Ai |−1) all equal the same polynomial, say g(t).
P
Since i 2(|Ai | − 1) = 2k, it has the unique solution
Y
Ri (t) = (2k + 2)(2k + 1) (t − āj )2(|Aj |−1) ,
j6=i
with
g(t) = (2k + 2)(2k + 1)
Y
(t − āj )2(|Aj |−1) .
j∈[r]
Therefore the second system also has a unique solution, where the f¯i (t) are obtained by
integrating gi (t) twice with some specific integration constants. For a fixed i we can again
assume āi = 0. Then both integration constants have been zero for this i, hence f¯i (0) = 0
and f¯i0 (0) = 0. Since gi is non-negative and zero only at isolated points, f¯i is strictly convex,
hence non-negative and zero only at t = 0. Therefore Q̄i (t) is positive for t 6= 0. Since
we chosePāi = 0, we can quickly
the correspondence between the coefficients of
P compute
j
j
Q̄i (t) = j q̄i,j t and of Ri (t) = j ri,j t :
ri,j = 2|Ai |(2|Ai | − 1) + 4j|Ai | + j(j − 1) q̄i,j .
In particular
Q̄i (0) = q̄i,0 =
ri,0
Ri (0)
=
> 0,
2|Ai |(2|Ai | − 1)
2|Ai |(2|Ai | − 1)
therefore Q̄i (t) is everywhere positive, hence so is Qi (t) if one chooses Ii possibly even closer
to āi , since the solutions of linear equation systems move continuously when one deforms the
entries of the equation system by a homotopy (as long as the determinant stays non-zero),
since the determinant and taking the adjoint matrix are continuous maps. The positivity of
Qi (t) finishes the proof.
3. Projections of deformed products of simple polytopes
In the previous section, we saw an explicit construction of polytopes whose k-skeleton is
equivalent to that of a product of simplices. In this section, we provide another construction of
(k, n)-PPSN polytopes, using Sanyal & Ziegler’s technique of “projecting deformed products
of polygons” [Zie04, SZ09] and generalizing it to products of arbitrary simple polytopes.
This generalized technique consists in suitably projecting suitable polytopes combinatorially
equivalent to a given product of simple polytopes in such a way as to preserve its complete
k-skeleton. The special case of products of simplices then yields (k, n)-PPSN polytopes.
3.1. General situation. We first discuss the general setting: for any given product P =
P1 × · · · × Pr of simple polytopes, we construct a polytope P ∼ combinatorially equivalent
to P and whose k-skeleton is preserved under the projection on the first d coordinates.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
9
Deformed products of simple polytopes. Let P1 , . . . , Pr be simple polytopes of respective
dimensions n1 , . . . , nr and facet descriptions Pi = {x ∈ Rni | Ai x ≤ bi }, where each real matrix
Ai ∈ Rmi ×ni has one row for each of the mi facets of Pi and ni = dim Pi many columns, and
bi is P
a right-hand side vector in Rmi . The
P product P = P1 × · · · × Pr then has dimension
n := i∈[r] ni , and is given by the m := i∈[r] mi inequalities


 
A1
b1



.
.
..

 x ≤  ..  .
Ar
br
The left hand m × n matrix shall be denoted by A. It is proved in [AZ99] that for any
matrix A∼ obtained from A by arbitrarily changing the zero entries above the diagonal blocks,
there exists a right-hand side b∼ such that the deformed polytope P ∼ defined by the inequality
system A∼ x ≤ b∼ is combinatorially equivalent to P . The equivalence is the obvious one: it
maps the facet defined by the i-th row of A to the one given by the i-th row of A∼ , for all i.
Following [SZ09], we will use this “deformed product” construction in such a way that the
projection of P ∼ to the first d coordinates preserves its k-skeleton in the following sense.
Preserved faces and the Projection Lemma. For integers n > d, let π : Rn → Rd
denote the orthogonal projection to the first d coordinates, and τ : Rn → Rn−d denote
the dual orthogonal projection to the last n − d coordinates. Let P be a full-dimensional
simple polytope in Rn , with 0 in its interior. The following notion of preserved faces — see
Figure 2 — will be used extensively in the end of this paper:
Definition 3.1 ([Zie04]). A proper face F of a polytope P is strictly preserved under π if
(i) π(F ) is a face of π(P ),
(ii) F and π(F ) are combinatorially isomorphic, and
(iii) π −1 (π(F )) equals F .
s
q
r
p
s
q
p
r
(a)
(b)
Figure 2. (a) Projection of a tetrahedron onto R2 : the edge pq is strictly
preserved, while neither the edge qr, nor the face qrs, nor the edge qs are
(because of conditions (i), (ii) and (iii) respectively). (b) Projection of a
tetrahedron to R: only the vertex p is strictly preserved.
The characterization of strictly preserved faces of P uses the normal vectors of the facets
of P . Let F1 , . . . , Fm denote the facets of P and for all i ∈ [m], let fi denote the normal
10
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
vector of Fi , and gi = τ (fi ). For any face F T
of P , let ϕ(F ) denote the set of indices of the
facets of P containing F , i.e., such that F = i∈ϕ(F ) Fi .
Lemma 3.2 (Projection Lemma [AZ99, Zie04]). A face F of the polytope P is strictly preserved under the projection π if and only if {gi | i ∈ ϕ(F )} is positively spanning.
A first construction. Let t ∈ {0, 1, . . . , r} be maximal such that the matrices A1 , . . . , At
P
P
are entirely contained in the first d columns of A. Let m := ti=1 mi and n := ti=1 ni .
By changing bases appropriately, we can assume that the bottom ni × ni block of Ai is the
identity matrix, for each i ≥ t + 1. In order to simplify the exposition, we also assume first
that n = d, i.e., that the projection on the first d coordinates separates the first t block
matrices from the last r − t. See Figure 3a.
Let {g1 , . . . , gm } ⊂ Rn−d be a set of vectors such that G := {e1 , . . . , en−d } ∪ {g1 , . . . , gm } is
the Gale transform of a full-dimensional simplicial neighborly polytope Q — see [Zie95, Mat02]
for definition and properties of Gale duality. By elementary properties of the Gale transform,
Q has m + n − d vertices, and dim Q = (m + n − d) − (n − d) − 1 = m − 1. In particular,
m−1
every subset of b m−1
2 c vertices spans a face of Q, so every subset of m + n − d − b 2 c =: α
elements of G is positively spanning.
We deform the matrix A into the matrix A∼ of Figure 3a, using the vectors g1 , . . . , gm
to deform the top m rows. We denote by P ∼ the corresponding deformed product. We say
that a facet of P ∼ is “good” if the right part of the corresponding row of A∼ is covered by a
vector of G, and “bad” otherwise. Bad facets are hatched in Figure 3a. Observe that there
are β := m − m − n + d bad facets in total.
(a)
(b)
Figure 3. The deformed matrix A∼ (a) when the projection does not slice
any block (n = d), and (b) when the block At+1 is sliced (n < d). Horizontal
hatched boxes denote bad row vectors. The top right solid block is formed by
the vectors g1 , . . . , gm .
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
11
Let F be a k-face of P ∼ . Since P ∼ is a simple n-dimensional polytope, F is the intersection
of n − k facets, among which at least γ := n − k − β are good facets. If the corresponding
elements of G are positively spanning, then F is strictly preserved under projecting on the
first d coordinates. Since we have seen that any subset of α vectors of G is positively spanning,
F will surely be preserved if α ≤ γ, which is equivalent to
m−1
k ≤ n−m+
.
2
Thus, under this assumption, we obtain a d-dimensional polytope whose k-skeleton is combinatorially equivalent to the one of P = P1 × · · · × Pr .
When the projection slices a block. We now discuss the case when n < d, for which the
method is very similar. We consider vectors g1 , . . . , gm+d−n such that G := {e1 , . . . , en−d } ∪
{g1 , . . . , gm+d−n } is the Gale dual of a neighborly polytope. We deform the matrix A into the
matrix A∼ shown in Figure 3b, using again the vectors g1 , . . . , gm to deform the top m rows
and the vectors gm+1 . . . , gm+d−n to deform the top d − n rows of the nt+1 × nt+1 bottom
identity submatrix of At+1 . This is indeed a valid deformation since we can prescribe the
nt+1 × nt+1 bottom submatrix of At+1 to be any upper triangular matrix, up to changing the
basis appropriately. For the same reasons as before,
elements of G is positively spanning;
(1) any subset of at least α := m+n−n− m+d−n−1
2
(2) the number of bad facets is β := m−m−n+n, and thus any k-face of P ∼ is contained
in at least γ := n − k − β good facets.
Thus, the condition α ≤ γ translates to
m+d−n−1
k ≤ n−m+
,
2
and we obtain the following proposition:
Proposition 3.3. Let P1 , . . . , Pr be simple polytopes of respective dimension
ni , and with mi
P
many facets. For a fixed integer d ≤ n, let t be maximal such that ti=1 ni ≤ d. Then there
exists a d-dimensional polytope whose k-skeleton is combinatorially equivalent to that of the
product P1 × · · · × Pr as soon as
$
!%
r
t
X
X
1
0 ≤ k ≤
(ni − mi ) +
d−1+
(mi − ni )
.
2
i=1
i=1
In the next two paragraphs, we present two improvements on the bound of this proposition.
Both use colorings of the graphs of the polar polytopes Pi4 , in order to weaken the condition
α ≤ γ, in two different directions:
(i) the first improvement decreases the number of required vectors in the Gale transform G,
which in turn, decreases the value of α;
(ii) the second one decreases the number of bad facets, and thus, increases the value of γ.
12
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
Multiple vectors. In order to raise our bound on k, we can save vectors of G by repeating
some of them several times. Namely, any two facets that have no k-face in common can share
the same vector gj . Since any two facets of a simple polytope containing a common k-face
share a ridge, this condition can be expressed in terms of incidences in the graph of the polar
polytope: facets not connected by an edge in this graph can use the same vector gj . For a
graph H, we denote as usual its chromatic number by χ(H). Then, each Pi with i ≤ t only
contributes χi := χ(sk1 Pi4 ) different vectors in G, instead of mi of them. Thus, we only need
P
in total χ = ti=1 χi different vectors gj . This improvement replaces m by χ in the formula
of α, while β and γ do not change, and the condition α ≤ γ is equivalent to
χ−d−n−1
.
k ≤ n−m+m−χ+
2
Thus, we obtain the following improved proposition:
Proposition 3.4. Let P1 , . . . , Pr be simple polytopes. For each polytope Pi , denote by ni
its dimension, by mi its number of facets, and by χi := χ(sk1 Pi4 ) the chromatic number of
the graph of its polar polytope Pi4 . For a fixed integer d ≤ n, let t be maximal such that
Pt
i=1 ni ≤ d. Then there exists a d-dimensional polytope whose k-skeleton is combinatorially
equivalent to that of the product P1 × · · · × Pr as soon as
$
!%
r
t
t
X
X
X
1
0 ≤ k ≤
(ni − mi ) +
(mi − χi ) +
d−1+
(χi − ni )
2
i=1
i=1
i=1
Example 3.5. Since polars of simple polytopes are simplicial, χi ≥ ni is an obvious lower
bound for the chromatic number of the dual graph of Pi . Polytopes that attain this lower
bound with equality are characterized by the property that all their 2-dimensional faces have
an even number of vertices, and are called even polytopes.
If all Pi are even polytopes, then n = χ, and we obtain a d-dimensional polytope with the
same k-skeleton as P1 × · · · × Pr as soon as
d−1
k ≤ n−m+m−n+
.
2
In order to maximize k, we should maximize m − n, subject to the condition n ≤ d. For
example, if all ni are equal, this amounts to ordering the Pi by decreasing number of facets.
Scaling blocks. We can also apply colorings to the blocks Ai with i ≥ t + 1, by filling in the
area below G and above the diagonal blocks. To explain this, assume for the moment that
χi ≤ ni+1 for a certain fixed i ≥ t + 2. Assume that the rows of Ai are colored with χi colors
using a valid coloring c : [mi ] → [χi ] of the graph of the polar polytope Pi4 . Let Γ be the
incidence matrix of c, defined by Γj,k = 1 if c(j) = k, and Γj,k = 0 otherwise. Thus, Γ is a
matrix of size mi × χi . We put this matrix to the right of Ai and above Ai+1 as in Figure 4b,
so that we append the same unit vector to each row of Ai in the same color class. Moreover,
we scale all entries of the block Ai by a sufficiently small constant ε > 0.
In this setting, the situation is slightly different:
n−d hatched in Fig(1) In the Gale dual G,
P we do not need the ni basis vectors of R
ure 4b. Let a =
j<i nj denote the index of the last column vector of Ai−1 and
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
13
+
(a)
(b)
Figure 4. How to raise the dimension of the preserved skeleton by inserting
the incidence matrix Γ of a coloring of the graph of the polar polytope Pi4 .
Part (a) shows the situation before the insertion of Γ, and part (b) the changes
that have occurred. Bad row vectors and unnecessary columns are hatched.
The entries in the matrix to the left of Γ must be rescaled to retain a valid
inequality description of P .
P
b = 1 + j≤i nj denote the index of the first column vector of Ai+1 . We define
G := {e1 , . . . , ea−d , eb−d , . . . , en−d } ∪ {g1 , . . . , gm } to be the Gale transform of a simplicial neighborly polytope Q of dimension m − 1 − ni . As before, any subset of
i −1
α := m + n − n − ni − b m+d−n−n
c vectors of G positively spans Rn−d .
2
(2) “Bad” facets are defined as before, except that the top mi − ni rows of Ai are not bad
anymore, but all of the first mi+1 − ni+1 + χi rows of Ai+1 are now bad. Thus, the
net change in the number of bad rows is χi − mi + ni , so that any k-face is contained
in at least γ := 2n − k − m + m − n + mi − ni − χi good rows. Up to ε-entry elements,
the last n − d coordinates of these rows correspond to pairwise distinct elements of G.
Applying the same reasoning as above, the k-skeleton of P ∼ is strictly preserved under
projection to the first d coordinates as soon as α ≤ γ, which is equivalent to
m + d − n − ni − 1
k ≤ n − m + mi − χi +
.
2
Thus, we improve our bound on k as soon as
m + d − n − ni − 1
m+d−n−1
∆ := mi − χi +
−
> 0.
2
2
For example, this difference ∆ is big for polytopes whose polars have many vertices but a
small chromatic number.
Finally, observe that one can apply this “scaling” improvement even if χi > ni+1 (except
that it will perturb more than the two blocks Ai and Ai+1 ) and to more than one matrix Ai .
14
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
Please see the example in Figure 5. In this picture, the Γ blocks are incidence matrices of
colorings of the graphs of the polar polytopes. Call “diagonal entries” all entries on the
diagonal of the ni × ni bottom submatrix of a factor Ai . A column is unnecessary (hatched in
the picture) if its diagonal entry has a Γ block on the right and no Γ block above. Good rows
are those covered by a vector gj or a Γ block, together with the basis vectors whose diagonal
entry has no Γ block above (bad rows are hatched in the picture).
Figure 5. How to reduce the number of vectors in the Gale transform using
various coloring matrices of polar polytopes. Situations where χi > ni+1 can
be accommodated for as illustrated by the matrix Γ2 in the picture.
Example 3.6.
(1) If Pi is a segment, then ni = 1, mi = 2 and χi = 1, so that ∆ = 1 if
m is even and 0 otherwise. Iterating this, if Pi is an s-dimensional cube, then ∆ ' 2s .
This yields neighborly cubical polytopes — see [JZ00, JS07].
(2) If Pi is an even cycle, then ni = 2, mi = 2p and χi = 2, so that ∆ = 2p − 3. This
yields projected products of polygons — see [Zie04, SZ09].
In general, it is difficult to give the explicit ordering of the factors and choice of deformation
that will yield the largest possible value of k attainable by a concrete product P1 × · · · × Pr of
simple polytopes, and consequently to summarize this improvement by a precise proposition as
we did for our first improvement. However, this best value can clearly be found by optimizing
over the finite set of all possible orderings and types of deformation. Furthermore, we can be
much more explicit for products of simplices, as we detail in the next section.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
1
-1
15
1
-1
1
-1
1
- 1
1
- 1
Figure 6. How to obtain PPSN polytopes from a deformed product construction, when the number s of segment factors exceeds the target dimension d of
the projection.
3.2. Projection of deformed product of simplices. We are now ready to apply this
general construction to the particular case of products of simplices. For this, we represent
the simplex ∆ni by the inequality system Ai x ≤ bi , where


−1 . . . −1
1



Ai := 

..


.
1
and bi is a suitable right-hand side. We express the results of the construction with a case
distinction according to the number s := |{i ∈ [r] | ni = 1}| of segments in the product 4n .
Proposition 3.7. Let n = (n1 , . . . , nr ) with 1 = n1 = · · · = ns < ns+1 ≤ · · · ≤ nr . Then
(1) for any 0 ≤ d ≤ s − 1, there exists a d-dimensional (k, n)-PPSN polytope as soon as
d
k ≤
− r + s − 1.
2
(2) for any s ≤ d ≤ n, there exists a d-dimensional (k, n)-PPSN polytope as soon as
d+t−s
k ≤
− r + s.
2
P
where t ∈ {s, . . . , r} denotes the maximal integer such that ti=1 ni ≤ d.
Proof of (1). This is a special case of the results obtainable with the methods of Section 3.1.
The best construction is obtained using the matrix in Figure 6, from which we read off that
16
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
(1) any subset of at least α := n − d2 vectors in G is positively spanning;
(2) the number of bad facets is β := r − s + 1, and therefore any k-face of P ∼ is contained
in at least γ := n − k − r + s − 1 good facets.
From this, the claim follows.
Proof of (2). Consider the deformed product of Figure 7a. Using similar calculations as
before, we deduce that
(1) any subset of at least α := t − s + n − d+t−s−1
vectors in G is positively spanning;
2
(2) the number of bad facets is β := r − t, and therefore any k-face of P ∼ is contained in
at least γ := n − k − r + t good facets.
This yields a bound of
d+t−s−1
− r + s.
k ≤
2
We optimize the final ‘−1’ away by suitably deforming the matrix At+1 as in Figure 7b.
This amounts to adding one more vector g? to the Gale diagram, so that the first row of At+1
ceases to be a bad facet. This deformation is valid because:
(1) the matrix


−1 . . . −1 ? . . . ?
M



..


.




M




1




..


.
1
still defines a simplex, as long as the ‘?’ entries are negative and M 0 is chosen
sufficiently large.
(2) we can in fact choose the new vector g? to have only negative entries, by imposing an
additional restriction on the Gale diagram G = {e1 , . . . , en−d , g1 , . . . , gd+t , g? } of Q.
Namely, we require the vertices of the (d + t)-dimensional simplicial polytope Q that
correspond to the Gale vectors g1 , . . . , gd+t to lie on a facet. This forces the remaining
vectors e1 , . . . , en−d , g? to be positively spanning, so that g? has only negative entries.
Finally, we reformulate Proposition 3.7 to express, in terms of k and n = (n1 , . . . , nr ), what
dimensions a (k, n)-PPSN polytope can have. This yields upper bounds on δpr (k, n).
Theorem 3.8. For any k ≥ 0 and n = (n1 , . . . , nr ) with 1 = n1 = · · · = ns < ns+1 ≤ · · · ≤ nr ,


2(k + r) − s − t if 3s ≤ 2k + 2r,
δpr (k, n) ≤
2(k + r − s) + 1 if 3s = 2k + 2r + 1,


2(k + r − s + 1) if 3s ≥ 2k + 2r + 2,
where t ∈ {s, . . . , r} is maximal such that
3s +
t
X
i=s+1
(ni + 1) ≤ 2k + 2r.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
17
1
-1
1
-1
1
-1
1
-1
-1-1 -1 -1 -1
-1-1 -1 -1 -1
M
M
M
(a)
(b)
Figure 7. Obtaining PPSN polytopes from a deformed product construction,
when few of the factors are segments. Part (a) shows the technique used so
far, and part (b) an additional optimization that exchanges a bad facet for a
new vector in the Gale transform.
Proof. Apply part (1) of Proposition 3.7 when 3s ≥ 2k + 2r + 2 and part (2) otherwise.
Remark 3.9. When all the ni ’s are large compared to k, the dimension of the (k, n)-PPSN
polytope provided by this theorem is bigger than the dimension 2k + r + 1 of the (k, n)-PPSN
polytope obtained by the Minkowski sum of cyclic polytopes of Theorem 2.6. However, if we
have many segments (neighborly cubical polytopes), or more generally if many ni ’s are small
compared to k, this construction provides our best examples of PPSN polytopes.
4. Topological Obstructions
In this section, we give lower bounds on the minimal dimension δpr (k, n) of a (k, n)-PPSN
polytope, applying and extending a method developed by Sanyal [San09] to bound the number
of vertices of Minkowski sums of polytopes. This method provides lower bounds on the target
dimension of any linear projection that preserves a given set of faces of a polytope. It uses
Gale duality to associate a certain simplicial complex K to the set of faces that are preserved
under the projection. Then lower bounds on the embeddability dimension of K transfer to
lower bounds on the target dimension of the projection. In turn, the embeddability dimension
is bounded via colorings of the Kneser graph of the system of minimal non-faces of K, using
Sarkaria’s Embeddability Theorem.
For the convenience of the reader, we first quickly recall this embeddability criterion. We
then provide a brief overview of Sanyal’s method before applying it to obtain lower bounds
on the dimension of (k, n)-PPSN polytopes. As mentioned in the introduction, these bounds
match the upper bounds obtained from our different constructions for a wide range of parameters, and thus give the exact value of the minimal dimension of a PPSN polytope.
18
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
4.1. Sarkaria’s embeddability criterion.
4.1.1. Kneser graphs. Recall that a k-coloring of a graph G = (V, E) is a map c : V → [k]
such that c(u) 6= c(v) for (u, v) ∈ E. As usual, we denote χ(G) the chromatic number of G
(i.e., the minimal k such that G admits a k-coloring). We are interested in the chromatic
number of so-called Kneser graphs.
Let Z be a subset of the power set 2[n] of [n]. The Kneser graph on Z, denoted KG(Z), is
the graph with vertex set Z, where two vertices X, Y ∈ Z are related if and only if X ∩ Y = ∅:
KG(Z) = Z, {(X, Y ) ∈ Z 2 | X ∩ Y = ∅} .
Let KGkn = KG [n]
denote the Kneser graph on the set of k-tuples of [n]. For example,
k
the graph KG1n is the complete graph Kn (of chromatic number n) and the graph KG25 is the
Petersen graph (of chromatic number 3).
Remark 4.1.
(1) If n ≤ 2k − 1, then any two k-tuples of [n] intersect and the Kneser
k
graph KGn is independent (i.e., it has no edge). Thus its chromatic number is
χ(KGkn ) = 1.
(2) If n ≥ 2k − 1, then χ(KGkn ) ≤ n − 2k + 2. Indeed, the map c : [n]
k → [n − 2k + 2]
defined by c(F ) = min(F ∪ {n − 2k + 2}) is a (n − 2k + 2)-coloring of KGkn .
In fact, it turns out that this upper bound is the exact chromatic number of the Kneser
graph: χ(KGkn ) = max{1, n − 2k + 2}. This result has been conjectured by Kneser in 1955,
and proved by Lovász in 1978 applying the Borsuk-Ulam Theorem — see [Mat03] for more
details. However, we will only need the upper bound for the topological obstruction.
4.1.2. Sarkaria’s Theorem. Our lower bounds on the dimension of (k, n)-PPSN polytopes rely
on lower bounds for the dimension in which certain simplicial complexes can be embedded.
Among other possible methods [Mat03], we use Sarkaria’s Coloring and Embedding Theorem.
We associate to any simplicial complex K the set system Z of minimal non-faces of K, that
is, the inclusion-minimal sets of 2V (K) r K. For example,
the complex of minimal non-faces of
[n+1]
the k-skeleton of the n-dimensional simplex is k+2 . Sarkaria’s Theorem provides a lower
bound on the dimension into which K can be embedded, in terms of the chromatic number
of the Kneser graph of Z.
Theorem 4.2 (Sarkaria’s Theorem). Let K be a simplicial complex embeddable in Rd , Z be
the system of minimal non-faces of K, and KG(Z) be the Kneser graph on Z. Then
d ≥ |V (K)| − χ(KG(Z)) − 1.
In other words, we get large lower bounds on the possible embedding dimension of K when
we obtain colorings with few colors of the Kneser graph on the minimal non-faces of K. We
refer to the excellent treatment in [Mat03] for further details.
4.2. Sanyal’s topological obstruction method. For given integers n > d, we consider
the orthogonal projection π : Rn → Rd to the first d coordinates, and its dual projection
τ : Rn → Rn−d to the last n − d coordinates. Let P be a full-dimensional simple polytope
in Rn , with 0 in its interior, and assume that its vertices are strictly preserved under π.
Let F1 , . . . , Fm denote the facets of P , and for all i ∈ [m], let fi denote the normal vector of
Fi , and gi = τ (fi ). For any face FTof P , let ϕ(F ) denote the set of indices of the facets of P
containing F , i.e., such that F = i∈ϕ(F ) Fi .
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
19
Lemma 4.3 (Sanyal [San09]). The vector configuration G = {gi | i ∈ [m]} ⊂ Rn−d is the
Gale transform of the vertex set of a (full-dimensional) polytope Q of Rm−n+d−1 . Up to a
slight perturbation of the facets of P , we can even assume Q to be simplicial.
We will refer to the polytope Q as Sanyal’s projection polytope. The faces of this polytope
capture the key notion of strictly preserved faces of P — remember Definition 3.1. Indeed, the
Projection Lemma 3.2 ensures that for any face F of P strictly preserved by the projection π,
the set {gi | i ∈ ϕ(F )} is positively spanning, which implies by Gale duality that the set of
vertices {ai | i ∈ [m] r ϕ(F )} forms a face of Q.
Example 4.4. Let P be a triangular prism in 3-space that projects to a hexagon as in
Figure 8a, so that n = 3, d = 2 and m = 5. The vector configuration G ⊂ R1 obtained by
projecting P ’s normal vectors consists of three vectors pointing up and two pointing down,
so that Sanyal’s projection polytope Q is a bipyramid over a triangle. An edge Fi ∩ Fj of P
that is preserved under projection corresponds to the face [5] r {i, j} of Q. Notice that the
six faces of Q corresponding to the six edges of P that are preserved under projection (in
bold in Figure 8a) make up the entire boundary complex of the bipyramid Q.
1
1
2 3
4
3
2
3
1
4 5
4
5
2
5
(a)
(b)
Figure 8. (a) Projection of a triangular prism and (b) its associated projection polytope Q. The six faces of Q corresponding to the six edges of P
preserved under projection (bold) make up the entire boundary complex of Q.
Let F be a subset of the set of all strictly preserved faces of P under π. Define K to be
the simplicial complex induced by {[m] r ϕ(F ) | F ∈ F}.
Remark 4.5. Notice that not all non-empty faces of K correspond to non-empty faces in F:
in Example 4.4, if F consist of all strictly preserved edges, then K is the entire boundary
complex of Sanyal’s projection polytope Q, so that it contains the edge {2, 3}. But then the
complementary intersection of facets, F1 ∩ F4 ∩ F5 , does not correspond to any non-empty
face of P .
Since the set of vertices {ai | i ∈ [m] r ϕ(F )} forms a face of Q for any face F ∈ F, and
since Q is simplicial, K is a subcomplex of the face complex of Q ⊂ Rm−n+d−1 . In particular,
when K is not the entire boundary complex of Q, it embeds into Rm−n+d−2 by stereographic
projection (otherwise, it only embeds into Rm−n+d−1 , as happens in Example 4.4).
20
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
Thus, given the simple polytope P ⊂ Rn , and a set F of faces of P that we want to preserve
under projection, the study of the embeddability of the corresponding abstract simplicial
complex K provides lower bounds on the dimension d in which we can project P . We proceed
in the following way:
(1) we first choose our subset F of strictly preserved faces sufficiently simple to be understandable, and sufficiently large to provide an obstruction;
(2) we then understand the system Z of minimal non-faces of the simplicial complex K;
(3) finally, we find a suitable coloring of the Kneser graph on Z and apply Sarkaria’s
Theorem 4.2 to bound the dimension in which K can be embedded: a t-coloring of
KG(Z) ensures that K is not embeddable into |V (K)| − t − 2 = m − t − 2, which by
the previous paragraph bounds the dimension d from below as follows:
Theorem 4.6 (Sanyal [San09]). Let P be a simple polytope in Rn whose facets are in general
position, and let π : Rn → Rd be a projection. Let F be a subset of the set of all strictly
preserved faces of P under π, K be the simplicial complex induced by {[m] r ϕ(F ) | F ∈ F},
and let Z be its system of minimal non-faces. If the Kneser graph KG(Z) is t-colorable, then
(1) if K is not the entire boundary complex of the Sanyal polytope Q, then d ≥ n − t + 1;
(2) otherwise, d ≥ n − t.
In the remainder of this section, we apply Sanyal’s topological obstruction to our problem.
The hope was initially to extend it to bound the target dimension of a projection preserving
the k-skeleton of an arbitrary product of simple polytopes. However, the combinatorics
involved to deal with this general question turn out to be too complicated and we restrict
our attention to products of simplices. We obtain in this manner bounds on the minimal
dimension δpr (k, n) of a (k, n)-PPSN polytope.
4.3. Preserving the k-skeleton of a product of simplices. In this section, we understand
the abstract simplicial complex K corresponding to our problem, and describe its system of
minimal non-faces.
The facets of 4n are exactly the products
ψi,j = 4n1 × · · · × 4ni−1 × (4ni r {j}) × 4ni+1 × · · · × 4nr ,
for i ∈ [r] and j ∈ [ni + 1]. We identify the facet ψi,j with the element j ∈ [ni + 1] of the
disjoint union [n1 + 1] ] [n2 + 1] ] · · · ] [nr + 1].
Let F = F1 ×· · ·×Fr be a k-face of 4n . Then F is contained in a facet ψi,j of 4n if and only
if j ∈
/ Fi . Thus, the set of facets of 4n that do not contain F is exactly F1 ] · · · ] Fr . Consequently, if we want to preserve the k-skeleton of 4n , then the abstract simplicial complex K
we are interested in is induced by
X
(3)
F1 ] · · · ] Fr ∅ =
6 Fi ⊂ [ni + 1] for all i ∈ [r], and
(|Fi | − 1) = k .
i∈[r]
Remark 4.7. In contrast to the general case, when we want to preserve the complete
k-skeleton of a product of simplices, the complex K cannot be the entire boundary complex of the Sanyal polytope Q. In consequence, the better lower bound from part (1) of
Sanyal’s Theorem 4.6 always holds, and we always use it from now on without further notice.
To prove that K cannot cover the entire boundary complex of Q, observe that
X
X
dim Q = m − n + d − 1 =
(ni + 1) −
ni + d − 1 = r + d − 1,
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
21
while dim K = r +k −1 by (3). A necessary condition for K to be the entire boundary complex
of Q is that dim K = dim Q − 1, which translates to d = k + 1. Now suppose that the entire
k-skeleton of 4n is preserved under projection to dimension k + 1. Then the projections of
those k-faces are facets of π(4n ). Since any ridge of the projected polytope is contained in
exactly two facets, and the entire k-skeleton of 4n is preserved, we know that any (k − 1)-face
of 4n is also contained in exactly two k-faces. But this can only happen if k = n − 1, which
means n = d.
Observe again that K can be the entire boundary complex of Q if we do not preserve all
k-faces of 4n — see Example 4.4.
The following lemma gives a description of the minimal non-faces of K:
Lemma 4.8. The system of minimal non-faces of K is
Z = G1 ] · · · ] Gr |Gi | =
6 1 for all i ∈ [r], and
X
(|Gi | − 1) = k + 1 .
i | Gi 6=∅
Proof. A subset G = G1 ] · · · ] Gr of [n1 + 1] ] P
[n2 + 1] ] · · · ] [nr + 1] is a face of K when it
can be extended to a subset F1 ] · · · ] Fr with (|Fi | − 1) = k and ∅ 6= Fi ⊂ [ni + 1] for all
i ∈ [r], that is, when
X
X
k ≥ i ∈ [r] | Gi = ∅ +
(|Gi | − 1) =
(|Gi | − 1).
i | Gi 6=∅
i∈[r]
Thus, G is a non-face if and only if
X
(|Gi | − 1) ≥ k + 1.
i | Gi 6=∅
P
If i | Gi 6=∅ (|Gi | − 1) > k + 1, then removing any element provides a smaller non-face. If
there is an i such that |Gi | = 1, then removing the
P unique element of Gi provides a smaller
6 1 for
non-face. Thus, if G is a minimal non-face, then i | Gi 6=∅ (|Gi | − 1) = k + 1, and |Gi | =
all i ∈ [r].
Reciprocally, if G is a non-minimal non-face, then it is possible to remove one element
keeping a non-face. Let i ∈ [r] be such that we can remove one element from Gi , keeping a
non-face. Then, either |Gi | = 1, or
X
X
(|Gj | − 1) ≥ 1 + (|Gi | − 2) +
(|Gj | − 1) ≥ k + 2,
j | Gj 6=∅
j6=i | Gj 6=∅
since we keep a non-face.
4.4. Colorings of KG(Z). The next step consists in providing a suitable coloring for the
Kneser graph on the system Z of minimal non-faces of K. Let S := {i ∈ [r] | ni = 1} denote
the set of indices corresponding to the segments, and R := {i ∈ [r] | ni ≥ 2} the set of indices
corresponding to the non-segments in the product 4n . We first provide a coloring for two
extremal situations.
P
Theorem 4.9 (Topological obstruction for low-dimensional skeleta). If k ≤ i∈R ni2−2 ,
then the dimension of any (k, n)-PPSN polytope cannot be smaller than 2k + |R| + 1:
δpr (k, n) ≥ 2k + |R| + 1.
22
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
Proof. Let k1 , . . . , kr ∈ N be such that
X
ki = k
and
i∈[r]
(
ki = 0
0 ≤ ki ≤
ni −2
2
for i ∈ S;
for i ∈ R.
Observe that
P
(1) such a tuple exists since k ≤ i∈R ni2−2 .
(2) for any minimal non-face G = G1 ] · · · ] Gr of Z, there exists i ∈ [r] such that
|Gi | ≥ ki + 2. Indeed, if |Gi | ≤ ki + 1 for all i ∈ [r], then
X
X
X
ki = k,
ki ≤
(|Gi | − 1) ≤
k+1 =
i | Gi 6=∅
i | Gi 6=∅
i∈[r]
which is impossible.
i +1]
For all i ∈ [r], we fix a proper coloring γi : [n
→ [χi ] of the Kneser graph KGnkii+2
+1 , with
[ki +2]
χi = 1 color if i ∈ S and χi = ni − 2ki − 1 colors if i ∈ R — see Section 4.1.1. Then, we define
a coloring γ : Z → [χ1 ] ] · · · ] [χr ] of the Kneser graph on Z as follows. Let G = G1 ] · · · ] Gr
be a given minimal non-face of Z. We choose arbitrarily an i ∈ [r] such that |Gi | ≥ ki + 2,
and again arbitrarily a subset g of Gi with ki + 2 elements. We color G with the color of g in
KGknii+2
+1 , that is, we define γ(G) = γi (g).
The coloring γ is a proper coloring of the Kneser graph KG(Z). Indeed, let G = G1 ]· · ·]Gr
and H = H1 ] · · · ] Hr be two minimal non-faces of Z related by an edge in KG(Z), which
means that they do not intersect. Let i ∈ [r] and g ⊂ Gi be such that we have colored G
with γi (g), and similarly j ∈ [r] and h ⊂ Gj be such that we have colored H with γj (h).
Since the color sets of γ1 , . . . , γr are disjoint, the non-faces G and H can receive the same
color γi (G) = γj (H) only if i = j and g and h are not related by an edge in KGknii+2
+1 , which
implies that g ∩ h 6= ∅. But this cannot happen, because g ∩ h ⊂ Gi ∩ Hi , which is empty by
assumption. Thus, G and H get different colors.
P
This provides a proper coloring of KG(Z) with
χi colors. By Theorem 4.6 and Remark 4.7, we know that the dimension d of the projection is at least
X
X
ni −
χi + 1 = 2k + |R| + 1.
i∈[r]
i∈[r]
Theorem 4.10 (Topological obstruction for high-dimensional skeleta). If k ≥
then any (k, n)-PPSN polytope is combinatorially equivalent to 4n :
X
δpr (k, n) ≥
ni .
i ni ,
1 P
2
Proof. Let G = G1 ] · · · ] Gr and H = H1 ] · · · ] Hr be two minimal non-faces of Z. Let
A = {i ∈ [r] | Gi 6= ∅ or Hi 6= ∅}. Then
X
X
X
(|Gi | + |Hi |) ≥
(|Gi | − 1) +
(|Hi | − 1) + |A|
i∈A
Gi 6=∅
= 2k + 2 + |A| >
Hi 6=∅
X
i∈[r]
ni + |A| ≥
X
(ni + 1).
i∈A
Thus, there exists i ∈ A such that |Gi | + |Hi | > ni + 1, which implies that Gi ∩ Hi 6= ∅, and
proves that G ∩ H 6= ∅.
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
23
Consequently, the Kneser graph KG(Z) is independent (and we can
P color it with only one
color). We obtain that the dimension d of the projection is at least
ni . In other words, in
this extremal case, there is no better (k, n)-PSN polytope than the product 4n itself.
P
Remark 4.11. Theorem 4.10 can sometimes be strengthened a little: If k = 21 Pni − 1, and
k + 1 is not representable as a sum of a subset of {n1 , . . . , nr }, then δpr (k, n) = ni .
Proof. As in the previous theorem, we prove that the Kneser graph KG(Z) is independent.
Indeed, assume that G = G1 ] · · · ] Gr and H = H1 ] · · · ] Hr are two minimal non-faces
of Z related by an edge in KG(Z). Then, G ∩ H is empty, which implies that for all i ∈ [r]
|Gi | + |Hi | ≤ ni + 1.
(4)
Let U = {i | Gi 6= ∅} and V = {i | Hi 6= ∅}. Then,
X
X
X
(|Gi | + |Hi |) =
(|Gi | − 1) +
(|Hi | − 1) + |U | + |V | = 2k + 2 + |U | + |V |
i∈U ∪V
i∈U
=
X
i∈V
(?)
ni + |U | + |V | ≥
X
ni + |U ∪ V | =
i∈U ∪V
i∈[r]
X
(ni + 1).
i∈U ∪V
Summing (4) over i ∈ U ∪ V implies that both the inequality (?) and (4) for i ∈ U ∪ V are
in fact equalities. The tightness of (?) implies furthermore that |U | + |V | = |U ∪ V |, so that
U ∩ V = ∅; in other words, Hi is empty whenever Gi is not. The equality in (4) then asserts
that |Gi | = ni + 1 for all i ∈ U , and therefore
X
X
k+1 =
(|Gi | − 1) =
ni
i∈U
i∈U
is representable as a sum of a subset of the ni , which contradicts the assumption.
Finally, to fill the gap in the ranges of k covered by Theorems 4.9 and 4.10, we merge both
coloring ideas as follows.
We partition [r] = A ] B and choose ki ≥ 0 for all i ∈ A and kB ≥ 0 such that
!
X
ki + kB ≤ k.
(5)
i∈A
P
We will see later what the best choice for A, B, kB and the ki ’s is. Let nB = i∈B ni . Color
kB +1
the Kneser graphs KGknii+2
with pairwise disjoint color sets with
+1 for i ∈ A and KGnB
(
ni − 2ki − 1 if 2ki ≤ ni − 2,
χi =
1
if 2ki ≥ ni − 2,
and


0
χB = nB − 2kB


1
if nB = 0,
if 2kB ≤ nB − 1,
if 2kB ≥ nB − 1,
colors respectively.
Observe now that for all minimal non-faces G = G1 ] · · · ] Gr ,
• either
P there is an i ∈ A such that |Gi | ≥ ki + 2,
• or i∈B | Gi 6=∅ (|Gi | − 1) ≥ kB + 1.
24
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
Indeed, otherwise
!
X
k+1 =
X
(|Gi | − 1) ≤
ki
+ kB ≤ k.
i∈A
i | Gi 6=∅
This permits us to define a coloring of KG(Z) in the following way. For each minimal non-face
G = G1 ] · · · ] Gr , we arbitrarily choose one of the following strategies:
(1) If we can find an i ∈ A such that |Gi | ≥ ki + 2, we choose an arbitrary subset g of Gi
with ki + 2 P
elements, and color G with the color of g in KGknii+2
+1 ;
(2) Otherwise, i∈B | Gi 6=∅ (|Gi | − 1) ≥ kB + 1, and we choose an arbitrary subset g of
]
]
(Gi r {ni + 1}) ⊂
[ni ]
i∈B
i∈B
with kB + 1 elements and color G with the color of g in KGknBB+1 .
By exactly the same argument as in the proof of Theorem 4.9, one can verify that this provides
a valid coloring of the Kneser graph KG(Z) with
X
χ := χ(A, B, ki , kB ) :=
χi + χB
i∈A
many colors. Therefore Sanyal’s Theorem 4.6 and Remark 4.7 yield the following lower bound
on the dimension d of any (k, n)-PPSN polytope:
X
d ≥ dk := dk (A, B, ki , kB ) :=
ni + 1 − χ ≥ δpr (k, n).
i
It remains to choose parameters A, B, and {ki | i ∈ A} and kB that maximize this bound.
We proceed algorithmically, by first fixing A and B, and choosing the ki ’s and kB to maximize
the bound on the dimension dk . For this, we first start with ki = 0 for all i and kB = 0, and
observe the variation of dk as we increase individual ki ’s or kB . By (5), we are only allowed
a total of k such increases. During this process, we will always maintain the conditions
2ki ≤ ni − 1 for all i ∈ Ai and 2kB ≤ nB (which makes sense by the formulas for χi and χB ).
We start with ki = 0 for all i and kB = 0. Then
X
χ(A, B, 0, 0) =
(ni − 1) + |S ∩ A| + nB
i∈A
=
X
i∈A
ni − |A| + |S ∩ A| +
X
i∈B
ni =
X
ni − r + |B ∪ S|,
i∈[r]
and
dk (A, B, 0, 0) = 1 + r − |B ∪ S|,
where S = {i ∈ [r] | ni = 1} denotes the set of segments.
We now study the variation of dk as we increase each of the ki ’s and kB by one. For i ∈ A,
increasing ki by one decreases χi by


2, if 2ki ≤ ni − 4,
1, if 2ki = ni − 3,


0, if 2ki ≥ ni − 2,
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
25
and hence increases dk by the same amount. Observe in particular that dk remains invariant
if we increase ki for some segment i ∈ S (because ni = 1 for segments). Thus, it makes sense
to choose B to contain all segments. Similarly, increasing kB by one decreases χB by


2, if 2kB ≤ nB − 3,
1, if 2kB = nB − 2,


0, if 2kB ≥ nB − 1,
and increases dk by the same amount.
Recall that we are allowed at most k increases of ki ’s or kB by (5). Heuristically, it
seems reasonable to first increase the ki ’s or kB that increase dk by two, and then these that
increase dk by one. Hence we get a case distinction on k, which also depends on A and B:
Theorem 4.12 (Topological obstruction, general case). Let k ≥ 0 and n = (n1 , . . . , nr ) with
r ≥ 1 and ni ≥ 1 for all i. Let [r] = A]B be a partition of [r] with B ⊃ S := {i ∈ [r] | ni = 1}.
Define
X ni − 2 nB − 1
+ max 0,
,
K1 := K1 (A, B) :=
2
2
i∈A
(
1 if nB is even and non-zero,
K2 := K2 (A, B) := i ∈ A | ni is odd +
0 otherwise.
Then the following lower bounds hold for the dimension of a (k, n)-PPSN polytope:
(1) If 0 ≤ k ≤ K1 , then
δpr (k, n) ≥ r + 1 − |B| + 2k;
(2) If K1 ≤ k ≤ K1 + K2 , then δpr (k, n) ≥ r + 1 − |B| + K1 + k;
(3) If K1 + K2 ≤ k, then
δpr (k, n) ≥ r + 1 − |B| + 2K1 + K2 .
This theorem enables us to recover Theorem 4.9 and Theorem 4.10:
Corollary 4.13. Let k ≥ 0 and n = (n1 , . . . , nr ) with r ≥ 1 and ni ≥ 1 for all i, and define
S := {i ∈ [r] | ni = 1} and R := {i ∈ [r] | ni ≥ 2}. Then
(1) If
0 ≤ k ≤
X ni − 2 i∈R
2
|S| − 1
+ max 0,
,
2
then δpr (k,
2k + |R| + 1.
Pn) ≥
P
(2) If k ≥ 21 ni then δpr (k, n) ≥ i ni .
Proof. Take A = R and B = S for (1), and A = ∅ and B = [r] for (2).
4.5. Explicit lower bounds. There is an algorithm to explicitly choose in general the partitions [r] = A ] B which yields the best bounds in Theorem 4.12. Since this algorithm is
quite technical, we just present the best results we obtain with this topological obstruction.
We refer to [MMPP09] for further details.
P
We fix K1 = K1 (R, S) and define d0 = r + 1 − |S| and n = i∈[r] ni . The best lower
bound dk that we obtain with this coloring can be summarized explicitly by the following
case distinction — see Figure 9:
26
B. MATSCHKE, J. PFEIFLE, AND V. PILAUD
dk
dk
A
B
n
n
d0
d0
0
K1
dk
0
k
dk
C
K1
k
D
n
n
d0
d0
0
K1
k
0
K1
k
Figure 9. Four different situations for the lower bound.
A. When |S| is even and non-zero: The bound dk is first increasing by two for 0 ≤ k ≤ K1 .
Then for each odd ni ≥ 3 we get a block with a first increment by one and a second
incrementP
by two. Then all increments are one until we reach the trivial bound
dk = n = i∈[r] ni .
B. When |S| is odd: As in case A, except that the first block corresponding to an odd
ni ≥ 3 consists only of one increment by one.
C. When |S| = 0 and there is an odd ni : As in the cases A and B, except that the first
two blocks corresponding to odd ni ’s consists only of one increment. If there is only
one odd ni then all increments from K1 on are one until we reach the trivial bound.
D. When all ni are even: The bound dk is first increasing by two for 0 ≤ k ≤ K1 . The
next increment isPzero, and all further increments are one until we reach the trivial
bound dk = n = i∈[r] ni .
Remark 4.14. Remark 4.11 still provides a better bound for certain cases, as for example
when k = 2 and n = (4, 2).
4.6. Comparison with Rörig and Sanyal’s results. In [RS09], Rörig and Sanyal dealt
with the special case n1 = . . . = nr =: n and r ≥ 2. They obtained in their Theorem 4.5 the
PRODSIMPLICIAL-NEIGHBORLY POLYTOPES
27
following bound:


,
if 0 ≤ k ≤ r n−2
2k + r + 1,
2
n−2 n−1 1
δpr (k, (n, . . . , n)) ≥ k + 2 r(n − 1) + 1, if r 2 < k ≤ r 2 ,


< k ≤ rn,
α + r(n − 1) + 1,
if r n−1
2
k−rb n−1
2 c
where α =
. We compare this with the graphs C (if n is odd) and D (if n is
b n+2
2 c
even) of Figure
9. Their first case matches exactly with the bounds of this paper, since
K1 = r n−2
. Plugging in k = K1 into their first two cases yields the
bound if n is odd,
2
rsame
but a different one if n is even. If n is even then the difference is 2 . The bound in their
second case has slope one, that is, it increases by one if k increases by one, and the bound
in their third case has a much smaller slope. Hence the bounds of Section 4.5 are stronger,
especially around k ≈ rn
2 . In the case r = 1 both bounds are equal, because at k = K1 we
already reach the best possible bound rn.
Acknowledgements
We thank Bernardo González Merino for extensive and fruitful discussions on the material
presented here.
We are indebted to Thilo Rörig and Raman Sanyal for discussions and comments on the
subject of this paper, and their very careful reading of an earlier draft.
We are grateful to the Centre de Reçerca Matemàtica (CRM) and the organizers of the
i-Math Winter School DocCourse on Combinatorics and Geometry, held in the Spring of 2009
in Barcelona, for having given us the opportunity of working together during three months
in a very stimulating environment.
Finally, we would like to thank Michael Joswig for suggesting to look at even polytopes.
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Technische Universität Berlin, Germany
E-mail address: [email protected]
Universitat Politècnica de Catalunya, Spain
E-mail address: [email protected]
École Normale Supérieure de Paris, France
E-mail address: [email protected]